| L(s) = 1 | + (−2.18 − 1.59i)2-s + (0.309 + 0.951i)3-s + (1.64 + 5.06i)4-s + (0.336 + 2.21i)5-s + (0.836 − 2.57i)6-s + 0.470·7-s + (2.78 − 8.55i)8-s + (−0.809 + 0.587i)9-s + (2.78 − 5.37i)10-s + (2.57 + 1.87i)11-s + (−4.30 + 3.12i)12-s + (−0.455 + 0.331i)13-s + (−1.02 − 0.748i)14-s + (−1.99 + 1.00i)15-s + (−11.0 + 8.05i)16-s + (−0.527 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 1.12i)2-s + (0.178 + 0.549i)3-s + (0.822 + 2.53i)4-s + (0.150 + 0.988i)5-s + (0.341 − 1.05i)6-s + 0.177·7-s + (0.982 − 3.02i)8-s + (−0.269 + 0.195i)9-s + (0.879 − 1.69i)10-s + (0.776 + 0.563i)11-s + (−1.24 + 0.903i)12-s + (−0.126 + 0.0918i)13-s + (−0.275 − 0.199i)14-s + (−0.516 + 0.258i)15-s + (−2.77 + 2.01i)16-s + (−0.127 + 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.516036 + 0.0456038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.516036 + 0.0456038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-0.336 - 2.21i)T \) |
| good | 2 | \( 1 + (2.18 + 1.59i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 - 0.470T + 7T^{2} \) |
| 11 | \( 1 + (-2.57 - 1.87i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.455 - 0.331i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.527 - 1.62i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.15 + 3.55i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.83 - 1.33i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.57 + 7.91i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 5.17i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.825 - 0.600i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 + (1.37 + 4.21i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.17 + 6.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-10.4 + 7.56i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.102 + 0.0745i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.863 - 2.65i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.95 - 15.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.91 - 5.75i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.46 + 4.52i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.61 + 11.1i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.01 - 3.64i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.61 - 8.04i)T + (-78.4 + 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.76359870996940172127335376877, −13.22616322751963732878220003854, −11.69307309142033945433954130681, −11.13208719026265471666077291270, −9.964563366249223358187566048381, −9.382472355091529155670027352239, −8.026561762881620037922145775041, −6.82897089154161743614652082254, −3.83999394468772514139806867964, −2.33206772867143503342010671730,
1.30193379590582868871331673828, 5.32741251771318038482834534986, 6.55204703138270377280349323383, 7.77444727406043832790750618608, 8.734334656031553475303353197609, 9.420832140826529847468994114061, 10.85949819132874383355828628689, 12.24472206524706177516929189165, 13.86477892528609217399001130322, 14.70265289186354281720963673258
